[Paper Review] On the Minimum Area of Null Homotopies of Curves Traced Twice

Zipei Nie|arXiv (Cornell University)|Nov 29, 2014

Advanced Numerical Analysis Techniques2 references3 citations

TL;DR

This paper presents an efficient algorithm to compute the minimum area of a null homotopy for closed plane curves that divide the plane into finitely many regions. It constructs a curve γ such that the minimum area required to null-homotope 2·γ is less than ε times that of γ for any ε > 0, demonstrating that doubling a curve can drastically reduce filling area in the plane.

ABSTRACT

We provide an efficient algorithm to compute the minimum area of a homotopy between two closed plane curves, given that they divide the plane into finite number of regions. For any positive real number $\varepsilon>0$, we construct a closed plane curve $γ$ such that the minimum area of a null homotopy of $2\cdotγ$ is less than $\varepsilon$ times that of $γ$. We also establish a lower bound on how complex a desired closed curve has to be.

Motivation & Objective

• To develop an efficient algorithm for computing the minimum area of a null homotopy between two closed plane curves that divide the plane into finitely many regions.
• To investigate whether doubling a closed curve in the plane can result in a significantly smaller filling area than twice the original curve’s filling area.
• To establish a lower bound on the complexity of curves that exhibit this non-linear scaling behavior in homotopy area.
• To generalize the cancellation norm to a weighted version for use in algorithmic computation of homotopy areas.

Proposed method

• Introduces a weighted cancellation norm on words over a symmetric set without identity, generalizing prior cancellation norms.
• Defines a weighted cancellation distance between curves and proves its equivalence to the minimum homotopy area under certain conditions.
• Uses dynamic programming and folding techniques on word representations of curves to compute the weighted cancellation norm efficiently.
• Applies a recursive argument based on partitioning the double curve into segments to bound the norm of the square of a word.
• Employs a combinatorial averaging argument over rotation positions to find a segment where unpaired positions are minimized, enabling inductive bounds.
• Proves that for any ε > 0, there exists a closed curve γ such that the minimum null homotopy area of 2·γ is less than ε times that of γ.

Experimental results

Research questions

• RQ1 Can the minimum area of a null homotopy for a curve traced twice be significantly smaller than twice the area for the original curve in the plane?
• RQ2 Is there an efficient algorithm to compute the minimum homotopy area between two closed plane curves that divide the plane into finitely many regions?
• RQ3 What is the minimal complexity of a closed curve that exhibits sublinear scaling in homotopy area upon doubling?
• RQ4 How does the weighted cancellation norm relate to the geometric minimum homotopy area in the plane?
• RQ5 Can the non-additive behavior of filling areas under curve doubling be systematically constructed and quantified?

Key findings

• An efficient algorithm computes the minimum area of a homotopy between two closed plane curves dividing the plane into finitely many regions in polynomial time and space.
• For any ε > 0, a closed plane curve γ exists such that the minimum area of a null homotopy of 2·γ is less than ε times the area of a null homotopy of γ.
• The weighted cancellation norm provides a combinatorial invariant that exactly computes the minimum homotopy area for such curves.
• The construction of such a curve γ is inspired by a highly twisted simple closed curve on the real projective plane.
• A lower bound is established on the complexity of curves that exhibit the sublinear homotopy area scaling behavior under doubling.
• The paper proves that ||w|| ≤ (m+1)/2 · ||w²|| for words of length ≤ 2^m, showing that the norm of a word can grow sublinearly under doubling.